Well yes and no.. she enjoys typical TV like walking dead, game of thrones, dexter, etc. etc. and has an interesting place in her heart for BBC drama's ( ie. downton abbey ) that I don't always understand.

To that end I think the key is just a really good story with characters one can get drawn in by and isn't not completely over the top.. problem is the only zombie one I really know is HOTD, closest thing to thrones I have seen is SAO lately, dexter.. well there is monster and a few good ones but I think that might be a bit dark for starters, and BBC type historical drama I haven't seen alot in anime and thats why I am hoping that people here will have some good suggestions..

As for those of you with the jokes.. not exactly helpful answers but you did make me laugh

How about Dusk Maiden of Amnesia, Fairy Tale, Haibane Renmei, Hanasaku Iroha, Kids on the Slope, Kimi ni Todoke - From Me to You, Occult Academy, Otome Yokai Zakuro, The Mystic Archives of Dantalian, or Usagi Drop. All but one of those is currently listed on Crunchyroll. The first season of Kimi ni Todoke is much better than the second.

If she likes Dexter, I would totally suggest Deathnote. I haven't seen a whole lot of it, but you could maybe try Case Closed. I'm a fan of Penguindrum, that has some comedy and a little bit of the fate/ destiny kind of thing. Good Luck!

Ummmmm, I would start with a movie before you delve into any series. A series is always way more daunting to take on for an anime new-comer. Therefore, I suggest practically anything by Studio Ghibli if she is into art at all and especially if she likes stories that focus on characters instead of events. I would saw either Howls Moving Castle, Castle in the Sky, or Princess Mononoke are good choices. My favorite is Castle in the Sky, but if she already thinks "anime is for kids"....I would go with Mononoke or Howl instead.

Also, if she can stand reading subtitles I usually introduce people to anime through subbed series because the voice acting is usually better and if they don't mind reading subtitles chances are they're going to like the sound of another language.

This. The beauty of poetry and the intensity of a high-speed card game come together in a fantastic blend, the characters are all wonderful, and the whole thing looks gorgeous. It's a great example of something that you just wouldn't find outside of anime.

Of course, maybe you're just going to start by showing her some dubbed anime (in which case, I can't help you).